Training-Image Assisted History Matching of Complex Multiple-Point Statistical Facies Models: A Pattern-Based Approach

Abstract

Abstract Traditional geostatistical methods that rely on two-point statistics (i.e., variogram models) to describe the spatial variability in rock properties are not appropriate for representing the spatial continuity patterns associated with complex geologic objects (e.g., curvilinear fluvial patterns and turbidite systems). Modern geostatistical techniques are developed to simulate multiple-point statistical (MPS) patterns from a training image (TI), as a conceptual model of geologic continuity, to generate model realizations with complex geologic connectivity. A major difficulty in using MPS methods is related to conditioning the simulation results on nonlinear dynamic production data. We develop a pattern-based model calibration approach for conditioning MPS-based facies models on nonlinear flow data. The formulation begins with defining a minimization problem in which the mismatch between predicted and observed production data is minimized. Since facies distributions are discrete variables, we use a parameterization method to transform them into a small number of continuous parameters that can be updated using gradient-based minimization. For this purpose, k-SVD sparse dictionary learning is adopted to approximate the connectivity patterns in the distribution of facies models. Because of the approximation involved in the k-SVD parameterization, an additional step is introduced, after each iteration, to map the resulting continuous models to the TI. To implement the mapping, a local search template is used to scan the TI to find local discrete patterns with smallest distances from the corresponding local patterns in the continuous solution. This process is repeated for all the grid cells in the continuous solution and the resulting local patterns are stored. To estimate the facies type in each grid cell, the collected patterns that intersect with any given cell are used to find the facies type with the highest frequency and assign it to that cell. Once the discrete solution is identified, it is passed to the continuous minimization problem to serve as a regularization term, for the next update iteration. The process is repeated until the discrete solution provides an acceptable match to the data. Numerical experiments are presented to evaluate the performance of the proposed approach for facies calibration in complex fluvial systems. The results suggest that the developed method presents a promising pattern-based approach for integration of production data into MPS-based facies models.